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Chapter 8: Problem 31

What assumptions must hold true to use the \(t\) distribution to make a confidence interval for \(\mu\) ?

Short Answer

Expert verified
In order to use a t-distribution to make a confidence interval for \(\mu\) one must assume that the data is normally distributed, the samples are independent, the sample size is smaller than 30, and the population standard deviation is unknown.

Step by step solution

01

Identify Data Distribution

The first assumption is that the data within the population follows a normal distribution. This is a crucial assumption for the use of the t-distribution. If this condition doesn't hold, another statistical method might be more appropriate.
02

Determine the Sample Independence

The second assumption is that the samples are independent of each other. In other words, the occurrence of a specific value in one sample does not affect the occurrence of a specific value in another sample. This is important to provide an unbiased estimation.
03

Assess the Sample Size

The third assumption is regarding the sample size. It is commonly agreed that for a sample size larger than 30, the normal distribution can be used in place of the t-distribution. Therefore, when using the t-distribution, we generally assume that the sample size is smaller than 30.
04

Assume Standard Deviation is Unknown

The fourth assumption, and one of the key reasons the t-distribution is used when creating a confidence interval for \(\mu\), is when the population standard deviation is unknown. If the standard deviation was a known quantity, we could use a z-distribution instead.

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